Problem 47 - Contemporary Precalculus

Question

(a) Graph \(y=x\) and \(y=e^{\ln x}\) in separate viewing windows [or use a split- screen if your calculator has that feature]. For what values of \(x\) are the graphs identical? (b) Use the properties of logarithms to explain your answer in part (a).

Step-by-Step Solution

Answer

Answer: The graphs of both functions y=x and \(y=e^{\ln x}\) are identical for all values of x > 0.

Step 1: Graphing the Functions

To graph the functions, we can use a graphing calculator or a software like Desmos or GeoGebra. We will graph the two functions and analyze their behavior and the points of intersection.

Step 2: Comparison of the Graphs

After graphing the functions, we can see that both graphs intersect at every point along the line \(y = x\). This means that for every \(x\), there is an identical value of \(y\) for both functions. Therefore, the graphs are identical for all values of \(x\).

Step 3: Properties of Logarithms

Now, let's use properties of logarithms to explain our findings in part (a). One of the main properties of natural logarithms states that \(e^{\ln x} = x\) for all \(x > 0\). This is why the graph of the function \(y=e^{\ln x}\) is identical to \(y=x\) for all \(x > 0\). So, for all \(x > 0\), we have: \(y_1 = x\) \(y_2 = e^{\ln x}\) Since \(e^{\ln x} = x\), we can conclude that for all \(x > 0\), \(y_1 = y_2\). The graphs of the two functions are identical for all values of \(x > 0\).

Key Concepts

Graphing Functions

Graphing functions is a visual way of displaying the relationship between two variables, usually denoted as x (the independent variable) and y (the dependent variable). When graphing the first function in our exercise, which is the identity function \( y = x \), we are essentially plotting all the points where the value of y is equal to the value of x. The result is a straight line that passes through the origin and slopes upwards at a 45-degree angle.

For the exponential function \( y = e^{\ln x} \), we might expect something more complex, but due to the properties of logarithms and exponentials, this is in fact equivalent to graphing the identity function as well. The graph of an exponential function typically curves upwards more and more steeply as x increases. However, in this special case, the function simplifies to \( y = x \), showing that a confident understanding of functions goes hand in hand with understanding their properties.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a special logarithm with the base e, where e is an irrational and transcendental number approximately equal to 2.71828. It is widely used in mathematics, physics, and engineering because of its natural properties when describing growth processes or decay.

The key property of the natural logarithm that's used in the exercise is \( e^{\ln x} = x \) for all positive values of x. This identity holds because taking the natural logarithm of x and then exponentiating it essentially 'cancels' each other out. This is akin to 'undoing' the function, similar to how adding and then subtracting the same number leaves you unchanged. Understanding this relationship is fundamental when working with exponential and logarithmic functions in calculus and higher-level mathematics.

Exponential Functions

Exponential functions are mathematical expressions of the form \( f(x) = a^x \), where the base a is a constant, and the exponent x is a variable. These functions are characterized by rapid growth or decay, and their graphs are distinctive curves that change steeply and non-linearly. Exponential functions are particularly important for modeling real-world phenomena such as population growth, radioactive decay, and compound interest.

When the base is e, we have the natural exponential function \( f(x) = e^x \), which has significant importance due to its unique mathematical properties, like a constant rate of growth. Students often encounter the natural exponential function in various problems and exercises; understanding its behavior, such as its inverse relationship with the natural logarithm, is essential for mastering topics in calculus and differentials.